La lecture à portée de main
Découvre YouScribe en t'inscrivant gratuitement
Découvre YouScribe en t'inscrivant gratuitement
23
pages
Ebook
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Publié par
Nombre de lectures
25
Publié par
Nombre de lectures
25
t g
Z
1 +h ( ;g ) ()d ();KS
2 S M
+ ()
1 M C x2 M T Mx
t k:k M g T Mx
2kkxH(x;) := 2
2~ H 2
~
2L (M) ~
0
Z
1 8a2C (T M); (a) = a(x;)d (x;) :=h ; (a) i 2 ;~ ~ ~ ~ L (M)o ~
T M
(a) ~ a ~~
2~ = :~ ~
~! 0 ~
2 t S M :=fkk = 1g g S Mx
S M
( ) S M~ ~!0
Ananoinwhictyshob.measureTheamainwstraUniquetegyshoisthethofeprobabilitsamemeasureasertint[17]onlyexceptofthatmeasurewLeohaasvaeknotowheredtheealnegativwhithLiouvillewtheeaklybci.e.haoticmanifoldsbtropehaevior.Sinai1.thatIntraoductiononlyLettewb[4,egeoa)compact,isconnected,Moreowsemiclassical,thariemannianenmanifold.vFworFallthePreciselydesicmeasures.ertiessemiclassicaldicit,aso-calledthatthevts,thisoinerisdicitendoLiouvillewnotedcurvwithKaenormonp.lationtheu-ofaccume.givortenrestrbi.e.ycthegemetriclimitoNonnenmacvquanerontheirw.fact,Thewithgeow,desicature.osucwofofisertiessicaloer,vthaterispropmeasuretropicthenolmogoroisydenedisastthedesicHamiltoniansemiclassicalothewmanifoldscorrespcurvondingdesictogthestrongstudyoticevw,andofthemandtoitciatedshoassoaldistributionsonenofLyquencesnge.wThistoquanQuantitgyConjecturecorrespwhetherondsistomeasuretheleastclassicalnegativkineticInenergyusedinv-Sinaithetocaseopoficlassicaltheofabsenceatureofparticular,pedotenolmogorotial.tropAsyanpyresultobservheable,athisbquanctedtitgeoyofcancbentreclosequandesicstizedenerviaInpseudoorks,dierenandtialh,calculuseandbtheenquansemiclassicaltumgivop1eratorresultcooutrosorespondinginstancetoese-theisofathoksequencelodistributionewwherecalledwsemiclas-ismeasure.propvortionalonetowstheaPlancmeasurekaconstanytonatnedKthis,v-SinaiistroptheofLaplacehBeltramiinoparianeunderrgeoaotaoronactingforon.dooroofTeature.ature,curvgeo.oOuronmaineoconcernsatisesincthisaapropr(Anosotipropcleywillergobyethetomeasure)uppaspconsequence,24,canThiseiswnwnalmostthelatsequencesconexpergeapunotheermeasuretumvdicittopropLiouvilleyonAcstudy[21,the7].asymptoticphenomenonbknoehaasvior,quanasergosectionalytendsertto.emain,halleofconcerningtresultheouldfolloewinganswsequencetheoftumdistrEriobuyti[18],onsdetermine:theositivmeasurenonptheofsemiclassicalsurfaceornian(atRiemanforcompactofaeofature).case[2],thetharamanntheiolmogoroLaplacianentheyofderivnctionspruertiesfsemeigenmeasuresofmanifoldsisnegativertiescurvop1prInasymptoticshethewstudythateKWv-ct.enAbstrayRIVI?REanGABRIELsemiclassicalSisCEositivAThisSURFimpliesOptVEDsuppCURofYsemiclassicalELcannotVeNONPOSITIiORtoFclosedMEASURESdesic,SEMICLASSICALeigenfunctionsOFtheOPYaplacianENTRannotwhereoncOpatebonoundeddfromobinishighagyelo.-pseudosubsequendierenwtialwithophereratorKofcsymmorebtitativollower[8]oundsandthei.e.tropound,ofsatisesmeasuresbereerenupp3].RuelleIntheherofwhalfabymanifoldsbAnAnvaccumdesiculationopforoinmanifoldstnegativ(ascurvw1ot t(g ) (g ) t t
Z X
+h ( ;g ) ()d ()KS j
S M j
+ t (S M;g ;)j
Z d 1X (d 1)max+
h ( ;g ) ()d () :KS j 2S M j=1
1 t := lim log sup jd gjmax t! 1 2S Mt
+j
d 1
2
max
Z d 1X1 +
h ( ;g ) ()d ():KS j2 S M j=1
1M C
Z
1 +h ( ;g ) ()d ();KS
2 S M
+h ( ;g ) ()KS
1
U
theqsualeditourysuppifeigenfunctionsandanonlydifKolmoorquasimoismeasuretheouldLiouvillesurfmeasureertiesinytheandcasetoofLyapunovanunstableAnosotumvsoywbut[15].ationInofththiseRegardingpreviousatureinequaliteye,nonpthemeFiw.standardotropdenotedInthecalpinequalitositiv[ekLycapunoofvvexpyonenotsyofWthisistotheectInrespthenwithenofnegativyandcomplexitnonpthebsul[6].folloRegardingLtheseact,propriemannianerties,titheemain(2)resulttofthat,Ananoftharaman,RecallKopyoiscphshoandyNonnenmeacehercoherenwyasthetocitshoandwgeneralizedthat,sequencesfortrateaensemiclassical)measurereononholdsanSoAtlynonoqstoevonmanifold,whetheronedichaswtcurvestimagenthatlarger,knomeasuretheretvariansurfacesvcurv-inproanicetosurfacesandew11],oclearaetocaniatedincwassoTheoremerebcumonnentheeegativseecurvaturnonnw:asemiclassicise.Itdesicdenitions).Landvdetailsanore[19]wheresystemsmeforthatBisendixov-Sinaiappdorlarge[23]upp(seeonentymeasuretropthisenthatmetric)ofcalledass(alsocannotv-Sinaitomogorodesics.olthatKisthewithoutconstructedab.ishetheofmaximalErgoexpansionforrateofofprotheforgeoydesicconstructoquasimowcandatthesurfacefactsthefewergoathat'sinareentheOurpenositivsemiclassicalesequencesLytheapunotvareexprenoneninequalittsen[6].generalizedCompareddes)withetheDonnelly'soriginalcanresultlastfromthe[2],manifold:thisknoineLiouvilleqisualinottdesicyagivositiveresifanofexplicceitiouvillelobwresulterisbanounddenseonrthesubsetenoftropeyature).ofouraofsemicthelassicalpropmeasure.ofFoforositivinstance,curvfor[20,manifoldsitoecamefthatcronstantsnegativtebcurvadaptedature,thethiswingloaw:er1.1.betoundbcanabompecrctee,written(1)assurfacrecallofusositiveLetc.onalHoewletevber,aitalcanasurtuThen,rnooutgeothatundery.measuretroparianisnayvforeryassertslRuelleargeduequandynamicaltitwherytheoremandainalsothis.case,thethegorpreviousentrloanwyerenbhasoundthecanerbexpeatnegativointe.(whicparticular,hresultwwsouldtheimplyortthatanitsemiclisianmeasureemptbyreducedresult).closedComgeobiningWtheseunderlinetourwyoalsoobservtatitheonsdes[4],btheDonnellyyInw9],ereconsideredleadquestiontoQuanformUniqueulateditheyconjecturepacthat,etsfeigenfunctionsohervanthatythissemiclassicalquestion,measureouenan,exceptionaloneofhasdev-Sinaithateoncenwithonexample,partsathemeasure(evcaifrLiouvilleolmogoroisriedicKanddhabeyparticula1.1.zeroRIVI?REtropG..atheoremclosedthegeotrop2ofdesicmeasureswillforhaofvofeLaplacian.enthetropwysituationszeroslighTheydieatlsoouraskyedtheabtropout(ifthetoextensionuasimoofwthisbconjectureconsistentowithmanifoldsconstruction.withoutecmakonjaugaobservtoneassumptionsptheoinittsnot[4].wnIntheamearecenurewhilergooforositivforLiouvillegeosucothatonwsurfaceerenonpableetoatupro.vfact,ethethatustheirthecoanjisethancture,holdstheforestanwnyinsurfacedirectionwiththatanexistsAnosoopvandgeoindesicaoiwt(forinstancetpweorkmeasure[17],hweL gjU jU
uU ()
t (g )t
S M
Z T1+ u s ; () = lim U (g )ds;
T!+1T 0
+ ()
Z
uh ( ;g ) U ()d ():KS
S M
Z
1 uh ( ;g ) U ()d ():KS
2 S M
uU ()
thebilliwardsthe.[17]Ourfoliationspurptheoseallinlothisitarticle.isltoatprotionveethetheoremconstruction1.1.eOuroinstrategymainwillprecisebs.erthewillswillathmecrucialasolmogoroinme[17]for(and(3)alsoe[this4])losoerywhere).itextendisoprobablyofbtinesectiontositivtetorpro(andveasier)lforprotheneereaderthetoofharesultveestrategya,go[4]ocdadvisorunderstandingandofurfacetheoutmethofordsoundfromhathesettvwisoinreferencesbwhnotecouldreethesurfacesgeomethesetrainicecpsituationisisthesimpler.theWwilleonwillandfowcusproonthethe4]mainvdierencewhatsinandwreferethelemmasreadehretost[section17,preci4]Then,forhothebdetailssurfacesofuresevfolloeraltolemmas.quanTheendix,crucialquanobsabetroprvIationncerelyisforthatueastoinelythealsoAnosopv2case,[11surfacesls.ofononperositievterestingeouldcurv[14],aturemainhatagevformehacinontinuousofstableeandisunstableerywherefmostoliationsemark.andasknoouldcossibleonjugateresultpconjugateoInintsal.vTheseandproptertiesertieswnoerets).atultthetheheartyofe,theOrganizationproicle.ofswine[4,ey3,of17]curvandghwropewillwillevwerirewritingfofyfromthatoevbenlongifsimilartheaspropadyertieseofwthesewillstable/unstabletodirectionsforareofwweakpreciselyeroinfortosurfacesdieofdierennonpgesost.iwtivweecurvproature,sectiontheywillarethesucien[4]tadaptedtosettinganswnonpearInthewquestiontheofinAnanetharaman-Nonnenmaconherpressures.inthethisewresultseaklypressurecsomehaoticthesSinaie.twledgementiouldng.sInm[4Anan,tro-3,this17],tiothereencouragingwtheasnonpaeddynamical.quanhetityydiscussionswhicsubheweras20cruciallydetaused:astrhefunstablebJacobianwofathevgeotodesicinobw.wInithefromcaseTheofadasurfacesanofofnonpnewositivulationetcurvtature,functionothentegralethecanwinrtrooundducedenedanvanalogue(andofalit.evThisRquanOnetitalsoywhethercomeswfrombthepsttoudythisoftoJacwithoutobipeldsts.andfact,issurfacescalledsthehaunstableeRiccatistablesolutionunstableorwiFhergopropdic[20](andnotcourseanconjugate[5]oinatThepdict,yethatnotconanuitwofyregardingescapingdicultsystems.ishaotictruecymore[20].andInthistheoincasewofdosurfacesseewithoutyconjugateapofointhists,yF1.2.rofeartireInand2,Ma??ehagivvaesurvsho2wnsurfacesthatnonpthisequanaturetithighliytisprelertiesatedetoneedthemakupptheerofLyorkapunoAsvallexpdetailsonenthetofsat[17,pwoinutdeaklye[12].eryInandfact,eryfortoanwyalwefordoneresultthesethisarofier-inorks,vearianrefertreaderprobabilitthymmeasuretheonofsanaloguesomeanandobtaine,explainonewhichaspcouldtsa.e.donebwhethermowingdknotheoftquestionatheofraisesargumenmeasuresInmiclassical3,eesdraofaystropoutlineenthetheof.onint4,ulesexplainrewthismainoffromextensioncanwhereeThein[6].etoofectofrespositiviscurvthetupp.esecr5,Lyeapunowvsameexpaso[17]nderiveanestimatettheattumpFinallyoinintappwithw.recThankssometoonthetumBirkhofromergoanddicfactstheorem,outtheKRuellevinequalitenyycanAbknoets.thenwrewrittenlikastofolloiws:thankdicyergoNalinictharamanisinmeasureducingLiouvilletotheqofsonnnforameinstextendsretulrestrictifromtheto3ositivMEASUREScurvLASSICALsMICsSEIOFthankOPYrENTRmanAndhelalso,fultheablothiswject.erWbreferoundreadoftotheorem,1.1]canmorebierewritten3
M
: S M ! M
(x;) := x V = (x;) d
1 S Mx
Z 2 T S M
0c(t) = (a(t);b(t)) t2 ( ; ) S M c(0) = c (0) =Z
H K (Z) =r 0 b(0) =r b(0) a (0) d (Z)
r X ()H
2
T S M =H V
S M 18
= (x;)2S M X;Y 2T S M
hX;Yi :=g (d (X);d (Y )) +g (K (X);K (Y )); x x
g x Mx
M S M
0 0J"(t) +R( (t);J(t)) (t) = 0;
0R(X;Y )Z X Y Z J (t) =
r 0 J(t) (t)
1C c : [a;b]!M s ( ; )s
@c =c Y (t) = (c (t))0 s js=0@s
2s7!c (t) c M C Y (t)s
c Y (t)
0 0c 8s2 ( ; ) c M (t) t (t)s
(V;W ) T S M H V